Venn Diagrams and Cardinal Numbers
PURPOSE:
The purpose of this activity is to develop skills in the use of Venn diagrams and
the Cardinal Number Formula.
REQUISITES: This activity assumes that students are familiar with the use of Venn diagrams
to show set operations and that students understand the Cardinal Number
Formula.
Cardinal Number Formula:
For any two sets A and B
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
DIRECTIONS : Draw an appropriate Venn diagram and use the given information along with
the Cardinal Number Formula to fill in the number of elements in each region.
Example: n(U) = 43, n(A) = 25, n(A∩B) = 5, n(B') = 30
For two sets A and B, the Venn diagram consists of four regions:
Since n(A ∩ B) = 5, then region 3 has 5 elements.
Since n(A) = 25, then region 2 has 25-5=20 elements.
Since n(B') = 30, then region 1 has 30-20=10 elements.
Now, since the Universal set consists of all four regions, then region 4 has 43 – (10 + 20 +
5) = 8 elements. Hence, the final Venn diagram is:
Complete the following for any two sets A and B.
1.
n(A) = 8, n(B) = 14, n(A ∩ B) = 5, n(U) = 23
2.
n(A) = 19, n(B) = 13, n(A ∪ B) = 25, n(A′) = 11
3.
n(B) = 20, n(A ∩ B) = 16, n(A ∪ B) = 30, n(A ∩ B)′ = 17
4.
n(A′) = 25, n(B) = 28, n(A′ ∩ B′) = 40, n(A ∩ B) = 10
5.
n(U) = 40, n(B′) = 19, n(A ∪ B)′ = 8, n(A ∩ B) = 6
For any three sets X, Y, and Z, the Venn diagram consists of eight regions:
Use what you have learned to complete the following for any three sets X, Y, and Z.
6.
n(X) = 57, n(X ∩ Y) = 35, n(X ∩ Z) = 21, n(Y ∩ Z) = 25, n(X ∪ Y) = 81,
n(Z) = 49, n(Y)′ = 52, n(X ∩ Y ∩ Z) = 15
7.
n(X) = 15, n(X′ ∩ Y′ ∩ Z′) = 21, n(Y ∩ Z′) = 3, n(X ∩ Y ∩ Z) =5,
n(Y ∩ Z) = 13, n(X ∩ Y′) = 9, n(X ∩ Z) = 8, n(Y ∪ Z) = 32
8.
n(X) = 26, n(X ∩ Z′) = 19, n(Y ∩ Z′) = 21, n(X ∩ Y ∩ Z) = 4, n(Z) = 18,
n(U) = 56, n(Y ∩ Z) = 6, n(X ∩ Y) = 12.
Venn Diagram Extra Practice
n(X) = 57,
n(X ∩ Y ) = 35,
n(X ∩ Z) = 21,
n(Y ∩ Z) = 25,
n(X ∪ Y ) = 81, n(Z) = 49, n(Y ) = 52, n(X ∩ Y ∩ Z) = 15.
X
Y
Z
U
n(X) = 15,
n(X ∩ Y ∩ Z ) = 21,
n(Y ∩ Z ) = 3, n(X ∩ Y ∩ Z) = 5
n(Y ∩ Z) = 13, n(X ∩ Y ) = 9, n(X ∩ Z) = 8, n(Y ∪ Z) = 32,.
X
Y
Z
U
Venn Diagram Extra Practice Solution
n(X) = 57,
n(X ∩ Y ) = 35,
n(X ∩ Z) = 21,
n(Y ∩ Z) = 25,
n(X ∪ Y ) = 81, n(Z) = 49, n(Y ) = 52, n(X ∩ Y ∩ Z) = 15.
X
13
Y
20
15
14
15
6
10
18
Z
U
n(X) = 15,
n(X ∩ Y ∩ Z ) = 21,
n(Y ∩ Z ) = 3, n(X ∩ Y ∩ Z) = 5
n(Y ∩ Z) = 13, n(X ∩ Y ) = 9, n(X ∩ Z) = 8, n(Y ∪ Z) = 32,.
X
21
Y
1
6
2
5
3
8
13
Z
U
7. It was once said that Country-Western songs emphasize three basic themes: love, prison,
and trucks. A survey of the local Country-Western radio station produced the following
data:
12 songs about a truck driver who is in love while in prison
13 about a prisoner in love
28 about a person in love
18 about a truck driver in love
3 about a truck driver in prison who is not in love
2 about people in prison who are not in love and do not drive trucks
8 about people who are out of prison, are not in love, and do not drive trucks
16 about truck drivers who are not in prison
T
d
P
a
b
e
c
h
f
g
L
U
(a) How many songs were surveyed?
Find the number of songs about:
(b) truck drivers
(c) prisoners
(d) truck drivers in prison
(e) people not in prison
(f) people not in love